A mass M1 on a frictionless plane is connected to a support O by a spring of stiffness k. Mass M2 is supported by a string of length l from M1.
a. What are the forces acting on M1 and M2?
b. By a consideration of these forces, derive the equations of motion of M1 and M2 using the small angle approximation
c. For M1=M2, find the normal frequencies and the ratio of the amplitudes of each mass for each mode.
Constants:
M1 - mass on frictionless plane
M2 - mass at the end of the pendulum which is attached at its pivot to M1.
X1 is the displacement from equilibrium of mass M1
X2 is the displacement from equilibrium position of mass M2
θ is the angle of the pendulum from equilibrium position
l is the length of the pendulum
*normal frequencies refer to frequencies in the normal mode of vibration which use variables A=X1+X2 and B= X1-X2. The letters don't matter. X1 and X2 refer to the horizontal displacement of the two masses, X1 for the mass on the frictionless plane, X2 for the mass on the pendulum.